Problem 36 - Thomas Calculus

Question

Graph the function $$ f(x)=\left\\{\begin{array}{ll}{\lfloor x\rfloor,} & { x \geq 0} \\ {\lceil x\rceil,} & { x<0}\end{array}\right. $$ Why is \(f(x)\) called the integer part of \(x ?\)

Step-by-Step Solution

Answer

\( f(x) \) gives the closest integer version of \( x \) toward zero, hence the integer part.

Step 1: Understand the Function Components

We have a piecewise function: \( f(x) = \begin{cases} \lfloor x \rfloor & \text{if} \ x \geq 0 \ \lceil x \rceil & \text{if} \ x < 0 \end{cases} \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \lceil x \rceil \) represents the smallest integer greater than or equal to \( x \). This means that for non-negative \( x \), the function outputs the floored value, and for negative \( x \), it outputs the ceiled value.

Step 2: Determine Critical Points and Intervals

Identify the critical point where the function changes, which is at \( x = 0 \). For intervals, consider two cases: \( x \geq 0 \) where \( f(x) = \lfloor x \rfloor \) and \( x < 0 \) where \( f(x) = \lceil x \rceil \).

Step 3: Sketch the Graph for x ≥ 0

For \( x \geq 0 \), plot \( f(x) = \lfloor x \rfloor \). Start from \( x = 0 \) and move right. For whole numbers, \( f(x) \) stays constant over the interval \([n, n+1)\) and jumps to the next integer at \( x = n+1 \). For example, \( f(0) = 0 \), \( f(1) = 1 \), and so on. Present horizontal steps starting from the y-axis.

Step 4: Sketch the Graph for x < 0

For \( x < 0 \), plot \( f(x) = \lceil x \rceil \). Start from slightly below zero and move left. Here, since we are dealing with ceiling, \( f(x) \) increments at each integer, i.e., for \( -1 < x < 0 \), \( f(x) = 0 \); for \( -2 < x \leq -1 \), \( f(x) = -1 \), forming similar horizontal steps moving left.

Step 5: Analyze the Graph

The graph consists of step-like structures on both sides of the y-axis. For positive \( x \), it looks like steps climbing rightward, and for negative \( x \), it has steps descending leftward. This visualization reflects the function's properties of flooring on non-negative \( x \) and ceiling on negative \( x \).

Step 6: Explain the Name 'Integer Part'

\( f(x) \) is called the integer part of \( x \) because the function outputs the closest integer towards zero relative to \( x \). For \( x \geq 0 \), it trims down towards the greatest whole number not exceeding \( x \), and for \( x < 0 \), it rounds up towards the least integer not less than \( x \).

Key Concepts

Greatest Integer Function

The greatest integer function, often represented as \( \lfloor x \rfloor \), is a mathematical function that returns the largest integer less than or equal to a given number. It's sometimes referred to as the "floor" function. Imagine it as a way of "trimming" a number down to its nearest whole number without exceeding it.

• For example, \( \lfloor 3.7 \rfloor = 3 \).
• Even with negative values, this concept holds: \( \lfloor -2.3 \rfloor = -3 \).

Note that for positive numbers, the greatest integer function results in dropping any decimal, effectively moving towards zero on the number line side. This foundation is part of why the function is crucial in piecewise functions, specifically when dealing with non-negative input values.

Least Integer Function

The least integer function is denoted by \( \lceil x \rceil \) and is commonly known as the "ceiling" function. This function outputs the smallest integer greater than or equal to a number, essentially "rounding up" to the nearest whole number.

• For example, \( \lceil 4.1 \rceil = 5 \).
• When dealing with negative numbers, \( \lceil -4.6 \rceil = -4 \).

The least integer function becomes significantly useful in piecewise functions when the input values are negative, as it ensures any number, regardless of its non-integer part, becomes a whole number upward in the number line.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting multiple sub-functions across different domains within a single coordinate plane. Each sub-function within the piecewise definition covers a specific interval, requiring careful identification of these intervals and the transitions between them.

• In the case of the function \( f(x) = \begin{cases} \lfloor x \rfloor & \text{if} \ x \geq 0 \ \lceil x \rceil & \text{if} \ x < 0 \end{cases} \), two intervals are critical: \( x \geq 0 \) and \( x < 0 \).
• The key point to note here is the transition at \( x = 0 \), where the function's behavior shifts.

Graphing each segment involves recognizing these domains: for non-negative values, plot horizontal steps that reflect the greatest integer function, starting at \( x = 0 \). For negative values, plot similar steps based on the least integer function, beginning just below zero. These step functions continue indefinitely in both directions, presenting a unique staircase-like graph.

Integer Part of a Number

The integer part of a number describes a concept where a number is transformed into an integer based on its location relative to zero on the number line. For the function \( f(x) \) given in the problem, this means applying the floor function for non-negative \( x \) and the ceiling function for negative \( x \), thereby yielding the integer closest to zero.

• For \( x \geq 0 \), the integer part trims \( x \) down to \( \lfloor x \rfloor \).
• For \( x < 0 \), it rounds up to \( \lceil x \rceil \).

Thus, the integer part of any real number here accounts for the nature of its sign, effectively acting as a bridge between the number and its closest integer counterpart along the number line. This concept is key in understanding operations involving both the greatest and least integer functions.